Transcript THE ART OF RESEARCH - Queen's University

THE ART OF RESEARCH
2005 Herzberg Lecture
M. Ram Murty, FRSC
Queen’s Research Chair
Queen’s University
What is research?
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The art of research is really the art of
asking questions.
In our search for understanding, the
SOCRATIC method of questioning is the
way.
QUESTION
Socrates taught Plato that all ideas
must be examined and fundamental
questions must be asked for proper
understanding.
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Some basic questions seem to defy simple
answers.
One can enquire into the nature of
understanding itself.
But then, this would take us into philosophy.
What is 2 + 2 ?
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The engineer takes out a calculator and finds the answer is 3.999.
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The physicist runs an experiment and finds the answer is between
3.8 and 4.2.
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The mathematician says he doesn’t know but can show that the
answer exists.
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The philosopher asks for the meaning of the question.
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The accountant closes all doors and windows of the room and asks
everyone, ‘What would you like the answer to be?’
Some Famous Questions
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What
What
What
What
What
What
is
is
is
is
is
is
life?
time?
space?
light?
a number?
a knot?
The Eight-fold Way
How to ask `good questions’?
A good question is one that leads to new
discoveries.
We will present eight methods of generating
`good questions’.
1. SURVEY
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The survey method consists of two steps.
The first is to gather facts.
The second is to organize them.
Arrangement of ideas leads to
understanding.
What is missing is also revealed.
The Periodic Table
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Dimitri Mendeleev
organized the existing
knowledge of the
elements and was
surprised to find a
periodicity in the
properties of the
elements.
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In the process of
writing a student text
in chemistry,
Mendeleev decided to
gather all the facts
then known about the
elements and
organize them
according to atomic
weight.
The periodic table now sits as the
presiding deity in all chemistry labs.
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David Hilbert
organized 23
problems at the ICM
in 1900.
Hilbert Problems
The 7th problem led to the
development of transcendental
number theory
The 8th problem is the Riemann
hypothesis.
The 9th problem led to the
development of reciprocity laws.
The 10th problem led to the
development of logic and
diophantine set theory.
The 11th problem led to the arithmetic
theory of quadratic forms.
The 12th problem led to class field
theory.
Who wants to be a millionaire?
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The Clay Mathematical Institute is offering $1 million
(U.S.) for the solution of any of the following seven
problems.
P=NP
The Riemann Hypothesis
The Birch and Swinnerton-Dyer conjecture
The Poincare conjecture
The Hodge Conjecture
Navier-Stokes equations
Yang-Mills Theory
www.claymath.org
2. OBSERVATIONS
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Careful observations
lead to patterns and
patterns lead to the
question why?
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The Michelson-Morley
experiment showed
that there was no
need to postulate a
medium for the
transmission of light.
Archimedes
Archimedes and his bath
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Archimedes goes to
take a bath and
notices water is
displaced in
proportion to his
weight!
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He was so happy with
his discovery that he
forgot he was taking
a bath!!
3. CONJECTURES
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Careful observations lead to well-posed
conjectures.
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A conjecture acts like an inspiring muse.
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Let us consider Fermat’s Last `Theorem.’
Fermat’s Last Theorem
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In 1637, Pierre de
Fermat conjectured the
following.
Fermat’s marginal note
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Fermat was reading
Bachet’s translation of
the work of
Diophantus.
9+ 16 = 25
25 + 144 = 169
He wrote his famous marginal note:
+ 225 = 289
To split a cube into a sum of two64
cubes
 fourth
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or a fourth power into a sum of two
powers and in general an n-th power as a sum
of two n-th powers is impossible.
I have a truly marvellous proof of this but this
margin is too narrow to contain it.
Srinivasa Ramanujan
Ramanujan was not averse to
making extensive calculations on
his slate.
Ramanujan made the following
conjectures.
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t is multiplicative: t( mn )=t(m)t (n) whenever
m and n are coprime.
t satisfies a second order recurrence relation for
prime powers.
|t(p)|< p11/2
These are called the Ramanujan conjectures
formulated by him in 1916 and finally resolved in
1974 by Pierre Deligne.
4. RE-INTERPRETATION
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This method tries to examine what is
known from a new vantage point.
An excellent example is given by
gravitation.
Newton’s theory of gravitation was
inspired by Kepler’s careful
observations.
Isaac Newton
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Gravity is a force.
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F=Gm1m2/r2
Albert Einstein
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Gravity is curvature of
space.
Gravity as curvature
Light and gravitational field
Bending of light due to gravity
Perihelion of Mercury
Black Holes
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In 1938,
Chandrasekhar
predicted the
existence of black
holes as a
consequence of
relativity theory.
What is re-interpretation?
Unique Factorization Theorem
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Every natural number can be written as a
product of prime numbers uniquely.
For example, 12 = 2 X 2 X 3 etc.
Unique Factorization Revisited
Euler
The Riemann Zeta Function
5. ANALOGY
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When two theories are analogous, we try
to see if ideas in one theory have
analogous counterparts in the other
theory.
Zeta Function Analogies
The Langlands Program
E. Hecke
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This analogy signalled
a new beginning in
the theory of Lfunctions and
representation theory.
Harish-Chandra
R. P. Langlands
The Doppler Effect
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When a train
approaches you the
sound waves get
compressed.
Police Radar
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The police use the
doppler effect to
record speeding cars.
6. TRANSFER
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The idea here is to
transfer an idea from
one area of research
to another.
A good example is
given by the use of
the doppler effect in
weather prediction.
7. INDUCTION
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This is essentially the
method of
generalization.
A simple example is
given by the following
observations.
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13+23 = 9 = 32
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13 +23+33 = 36 = 62
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A general pattern?
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13 + 23 + … + n3 =
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[n(n+1)/2]2
The Theory of L-functions
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GL(1): Riemann zeta
function.
GL(2): Ramanujan
zeta function.
Building on these two
levels, Langlands
formulated the
general theory for
GL(n).
8. CONVERSE
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Whenever A implies B we may ask if B
implies A.
This is called the converse question.
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A good example occurs in physics.
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Electromagnetism
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An electric current
creates a magnetic field.
One may ask if the
converse is true.
Does a magnetic field
create an electric current?
Converse Theory
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We have seen that the Riemann zeta function and
Ramanujan’s Delta series have similar properties.
We also learned that Langlands showed that these zeta
functions arise from automorphic representations.
The question of whether all such objects arise from
automorphic representations is called converse theory.
Langlands proved a 2-dimensional reciprocity law.
New Directions
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Feynman diagrams
Knot theory
Zeta functions
Multiple zeta values
NUMBER THEORY
AND PHYSICS
SUMMARY
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SURVEY
OBSERVATIONS
CONJECTURES
RE-INTERPRETATION
ANALOGY
TRANSFER
INDUCTION
CONVERSE